(1)
The table below shows
all the whole numbers
as the base numbers &
the respective triangular numbers
generated from these.
Triangular Numbers Table
Base
number
Triangular
number
•••••
•
••
•••
••••
•••••
0
(0 x 1)/2 = 0
1
(1 x2 )/2 = 1
4
(4 x 5)/2 = 10
9
(9 x 10)/2 = 45
16
(16 x 17)/2 = 136
•
•
•
•
•
•
•
•
(2)
The tabulation below shows
all the sub-totals of
the numbers per (1).
0 = 0
0 + 1 = 1
0 + 1 + 10 = 11
0 + 1 + 10 + 45 = 56
0 + 1 + 19 + 45 + 136 = 192
•
•
•
•
(3)
The tabulation below shows
the numbers per (2)
multiplied by 60
& the resulting products
increased by
respective uneven numbers
starting from 1.
60(0) + 1 = 1
60(1) + 3 = 63
60(11) + 5 = 665
60(56) + 7 = 3367
60(192) + 9 = 11529
•
•
•
•
(4)
The tabulation below shows
all the sub-totals of
the numbers per (3),
resulting in all whole numbers
to the 6th. power.
1 = 1 = 16
1 + 63 = 64 = 26
1 + 63 + 665 = 729 = 36
1 + 63 + 665 + 3367 = 4096 = 46
1 + 63 + 665 + 3367 + 11529 = 15625 = 56
•
•
•
•
From the tabulation above
one can deduce that in general
n to the 6th. power is
the summation of triangular numbers
as shown below:
n4 =
n-1
Σ
k=0
n-1
[Σ
k=0
[30k2(k6 + 1) + (2k + 1)]
]